More complete discussion of the time-dependence of the non-static line element for the universe

In a previous article,(1) I have shown that a continuous transformation of matter into radiation, occurring throughout the universe, as postulated by the astrophysicists, would necessitate a nonstatic line element for the universe, and have shown that the non-static character thus introduced might provide an explanation of the red shift in the light from the extra-galactic nebulae. In the present article, I wish to discuss the form of dependence of the line element on the time more completely than was possible on the previous occasion. This is a matter of considerable importance, since changes in the approximations which must be introduced to obtain a usable result affect to quite a different extent the expressions for the relation between red shift and distance and for the rate of annihilation of matter. Indeed, the possibility arises of slight changes from the treatment previously given which would leave the theoretical relation between red shift and distance still approximately linear, as observationally found, and yet produce a very considerable change in the calculated rate for the annihilation of matter

The superficial density of matter at a liquid-vapor boundary

The distribution of matter within the transition layer between the two phases of a fluid system is considered. Approximate values are obtained for the superficial density of matter T, calculated with reference to the Gibbs surface of tension as the dividing surface between the phases

Consideration of the Gibbs theory of surface tension

The Gibbs theory of surface tension is discussed. Detailed consideration is given to the structure of transition layers between phases. This provides theoretical information, as to the magnitude of surface tension and as to the location of the surface of tension, which can be used in making applications of the Gibbs theory

The Effect of Droplet Size on Surface Tension

The effect of droplet size on surface tension is given theoretical consideration with the help of results of the Gibbs thermodynamic theory of capillarity and of previous results of the author as to the sign and magnitude of superficial densities. It is concluded that surface tension can be expected to decrease with decrease in droplet size over a wide range of circumstances. In addition, approximate figures are obtained for the rate at which such decreases may be expected. The decreases become significant for very small drops. The results are of interest in view of the important role of surface tension in determining the behavior of small droplets

Temperature equilibrium in a static gravitational field

In the case of a gravitating mass of perfect fluid which has come to thermodynamic equilibrium, it has previously been shown that the proper temperature T0 as measured by a local observer would depend in a definite manner on the gravitational potential at the point where the measurement is made. In the present article the conditions of thermal equilibrium are investigated in the case of a general static gravitational field which could correspond to a system containing solid as well as fluid parts. Writing the line element for the general static field in the form ds2=gijdxidxj+g44dt2 i,j=1,2,3, where the gij and g44 are independent of the time t it is shown that the dependence of proper temperature on position at thermal equilibrium is such as to make the quantity T0sqrt[g44] a constant throughout the system

Weak quantization

Quantization is called weak when a motion apparently allowed by the equation ∫pdq=nh, has less than the normal a-priori weight. It is believed that the deficiency in a-priori weight is taken over, either by neighboring classically allowed motions, or by neighboring strongly quantized motions when such are present in the region of the phase-space considered. Weak quantization is to be expected when uncertainties arise as to the period that should be used in determining the limits of the phase integral ∫pdq. Several cases are considered; (a) when the period is so long that there is considerable chance of interruption by a quantum transition; (b) when a system has two apparent periods, a long true period T and a short quasi-period θ; (c) when the periodicity is disturbed frequently in a fortuitous manner as by molecular collisions. In case (b), the tendency towards quantization with respect to T may be gradually replaced by quantization with respect to θ as T is lengthened, and then the probability of quantum transitions which correspond to quantization with respect to T is weakened while that of transitions related to θ is strengthened. This suggests the possibility that the strengthening of the probability of transitions related to a period θ may be accompanied by a strengthening of quantization with respect to that period

On the behavior of non-static models of the universe when the cosmological term is omitted

If the cosmological term in the equations of relativistic mechanics is set equal to zero, it has been shown by Einstein that a non-static model of the universe filled with a homogeneous distribution of incoherent matter would expand to a maximum volume and then start contracting. This, however, is a very special model of the universe filled with a highly simplified fluid, and subjected to changes which can be shown to be thermodynamically reversible; and it has recently been pointed out by one of the present authors that we can also expect a similar expansion to a maximum volume with much more general models of the universe allowing irreversible as well as reversible changes in the fluid filling the model. The present article gives a somewhat detailed analysis of the behavior of a wide class of non-static models of the universe when the cosmological term is set equal to zero, and shows that we may expect a continued succession of expansions and contractions without reference to the reversible or irreversible nature of the processes taking place in the fluid filling the model. The bearings of this finding on the problems of relativistic thermodynamics, which have already been treated by one of the present authors, are again noted

Using nonequilibrium fluctuation theorems to understand and correct errors in equilibrium and nonequilibrium discrete Langevin dynamics simulations

Common algorithms for computationally simulating Langevin dynamics must discretize the stochastic differential equations of motion. These resulting finite time step integrators necessarily have several practical issues in common: Microscopic reversibility is violated, the sampled stationary distribution differs from the desired equilibrium distribution, and the work accumulated in nonequilibrium simulations is not directly usable in estimators based on nonequilibrium work theorems. Here, we show that even with a time-independent Hamiltonian, finite time step Langevin integrators can be thought of as a driven, nonequilibrium physical process. Once an appropriate work-like quantity is defined -- here called the shadow work -- recently developed nonequilibrium fluctuation theorems can be used to measure or correct for the errors introduced by the use of finite time steps. In particular, we demonstrate that amending estimators based on nonequilibrium work theorems to include this shadow work removes the time step dependent error from estimates of free energies. We also quantify, for the first time, the magnitude of deviations between the sampled stationary distribution and the desired equilibrium distribution for equilibrium Langevin simulations of solvated systems of varying size. While these deviations can be large, they can be eliminated altogether by Metropolization or greatly diminished by small reductions in the time step. Through this connection with driven processes, further developments in nonequilibrium fluctuation theorems can provide additional analytical tools for dealing with errors in finite time step integrators.Comment: 11 pages, 4 figure

On the weight of heat and thermal equilibrium in general relativity

In accordance with the special theory of relativity all forms of energy, including heat, have inertia and hence in accordance with the equivalence principle also have weight. The purpose of the present article is to investigate the thermodynamic implications of the idea that heat has weight. In particular an investigation is made to see if a temperature gradient is a necessary accompaniment of thermal equilibrium in a gravitational field, in order to prevent the flow of heat from regions of higher to those of lower gravitational potential. A preliminary non-rigorous treatment of this problem is first given by attempting to modify the classical thermodynamics only to the extent of associating with each intrinsic quantity of energy an additional amount of potential gravitational energy. In this way an expression is obtained for increase in equilibrium temperature with decrease in gravitational potential which, however, could in any case only be correct as a first approximation in a weak gravitational field. A discussion of the uncertainties and lack of rigor of this preliminary treatment is then given and the necessity pointed out for a rigorous treatment based on the principles of general relativity. A rigorous relativistic treatment is then undertaken using the extension of thermodynamics to general relativity previously presented by the author. The system to be treated is taken as a static spherical distribution of perfect fluid which has come to gravitational and thermodynamic equilibrium. The principles of relativistic mechanics are first applied to such a system in order to obtain results needed in the later work. And it is then shown that these mechanical principles themselves are sufficient to determine the temperature distribution as a function of potential in the simple case of black-body radiation. The principles of relativistic thermodynamics are then applied to this same case of pure black-body radiation and the same expression for temperature as a function of potential obtained by the thermodynamic as by the mechanical treatment. This may be regarded as giving some measure of check on the validity of the proposed relativistic thermodynamics. Following this, a thermodynamic treatment is given for the temperature distribution in the more general case of matter and radiation and a result found which harmonizes with that for radiation alone. A treatment is then given to the distribution of a perfect monatomic gas in a gravitational field both on the assumption that the total number of atoms must remain constant and on the assumption of the ready interconvertibility of matter and radiation. In the latter case the same dependence of concentration on temperature is obtained as was found by Stern and by the author for the case of flat space-time. Using a system of coordinates such that the line element for the sphere of fluid takes the form ds2=-eu(dr2+r2dθ2+r2sin2θdφ2)+eνdt2 the general result for the relation between gravitational potential and equilibrium temperature T0 as measured by a local observer in proper coordinates can be given by the equation d lnT0/dr=-1/2dν/dr This equation reduces in the case of a weak field to that obtained by the preliminary non-rigorous treatment, and gives a very small change of temperature with position in fields of ordinary intensity. The result, however, is one of great theoretical interest, since constant temperature throughout any system which has come to thermal equilibrium has hitherto been regarded as an inescapable thermodynamic conclusion. It is also not out of the question that the effect might sometime be of experimental or observational importance